Let $\Omega$ be a bounded open subset of ${ℝ}^n$, $n>2$. In $\Omega$ we deduce the global differentiability result $$u\in H^2(\Omega ,{ℝ}^N)$$ for the solutions $u\in H^1(\Omega ,{ℝ}^n)$ of the Dirichlet problem $$u-g\in H_0^1(\Omega ,{ℝ}^N),-\sum _i{D}_i{a}^i(x,u,Du)=B_0(x,u,Du)$$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{\xi }$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)=e^{i{L}_{\xi }}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=(L_{\xi }-i){(L_{\xi }+i)}^{-1}$. We also show that the unitary group $U_{ℳ}\subset L²(ℳ,\tau )$ with the strong operator topology is not an embedded submanifold...

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H\times [0,+\infty )$, where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H\times \left\{0\right\}$. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty$ with smooth boundary.

Let M be a separable $C^{\infty }$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a $C^{\infty }$ function, or of a $C^{\infty }$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a $C^{\infty }$ function on the whole of M.

We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller $C_{\Pi }^k$-map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.

Differential forms on the Fréchet manifold $ℱ(S,M)$ of smooth functions on a compact $k$-dimensional manifold $S$ can be obtained in a natural way from pairs of differential forms on $M$ and $S$ by the hat pairing. Special cases are the transgression map ${\Omega }^p\left(M\right)\to {\Omega }^{p-k}\left(ℱ(S,M)\right)$ (hat pairing with a constant function) and the bar map ${\Omega }^p\left(M\right)\to {\Omega }^p\left(ℱ(S,M)\right)$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].